Derive a finite phasor sum and remove its singularities

1. Establish the optical mechanism

For N equally illuminated narrow slits, adjacent paths differ by q wavelengths and adjacent phases by 2πq. The total field in single-slit units is A=Σ from j=0 to N−1 exp(i2πjq). This finite sum is always defined, including when every phasor points in the same direction.

For noninteger q, the geometric-series identity yields A=(1−exp(i2πNq))/(1−exp(i2πq)). Factoring phase terms gives intensity [sin(Nπq)/sin(πq)]². At integer q the ratio appears singular, but the original sum is N and the intensity is N². A numerical implementation should use the finite sum or a valid limit rather than divide 0 by 0.

Near a principal maximum, the closest zeros are displaced by 1/N in q. More illuminated slits narrow principal peaks while leaving their separation of one in q unchanged. Since q=d sinθ/λ, changing physical spacing d instead changes the angular positions. Secondary maxima are not generally equal-height copies or uniformly spaced in angle.

The N² peak does not mean total power scales as N² over the entire pattern. Averaging intensity over one q period removes cross terms and gives N in single-slit units. Narrower peaks compensate for increased height. Finite slit widths add an envelope, and imperfect coherence or unequal amplitudes change the original ideal array.

2. Work the original evidence

For N=4 and q=0.25, the phasors are 1,i,−1,−i and sum to zero. At q=0 the field is 4 and intensity 16. The nearest zero is 0.25 away in q, and period-averaged intensity is 4.

3. Predict and investigate

How can peak intensity grow quadratically while period-averaged intensity grows linearly?

Sum real and imaginary components directly, compare with the geometric ratio away from its removable singularities and integrate a full q period to check the power scaling.

Primary mechanism context. Checked 31 October 2026. Original prose, records and diagrams; no copied source exercises or physical light-source operation. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.